I A math confusion in deriving the curl of magnetic field from Biot-Savart

AI Thread Summary
The discussion centers on a confusion regarding the derivation of the curl of the magnetic field from the Biot-Savart law as presented in Griffiths' "Introduction to Electrodynamics." A specific concern is raised about why the second term in equation 5.55 is zero, particularly when the current density J approaches infinity. The poster seeks clarification on how to demonstrate that the surface integral also results in zero under these conditions. The inquiry highlights a desire for a clear explanation to resolve this mathematical confusion. Overall, the thread emphasizes the need for understanding the behavior of integrals in electromagnetic theory.
Brian Tsai
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Why the surface integral is 0 even the J itself extends to infinity (as in the case of an infinite straight wire).
I am recently reading "Introduction to Electrodynamics, Forth Edition, David J. Griffiths " and have a problem with the derive of the curl of a magnetic field from Biot-Savart law. The images of pages (p.232~p233) are in the following:

螢幕擷取畫面 2023-04-03 133932.png

螢幕擷取畫面 2023-04-03 134140.png

The second term in 5.55(page 233) is 0. I had known the reason in case of that the current density declined to 0 on the surface. My question is how to prove the surface integral will also be 0 when J extends to infinite(red block).

P.S. : This is my first time asking a question in English, and I had done my best to decrease the improper use of English. I sincerely hope that anyone who notices my post can answer my confusion and don't be mad at my terrible use in English
 
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